A short presentation on Brenier-Monge-Ampère gravity given at the Tetrahedron VII workshop

1. Mathématiques - Informatique
Monge-Ampère Gravity
Bruno Lévy
ParMA project-team
Joint work with Yann Brenier, Pierre Boldrini and Roya Mohayaee
Inria Saclay
Labo. de Mathématiques d’Orsay
Université Paris Saclay
Tetrahedron VII
BSC-UPC
Barcelona
Oct 9-11 2023

2. Mysteries in the sky …

3. Mysteries in the sky
Vera Rubin - 1962

4. Mysteries in the sky
There is more mass than what we observe
Vera Rubin - 1962

5. There is more mass than what we observe

6. There is more mass than what we observe

7. Type Ia supernovae
“standard candles”
Permutter
Riess

8. Type Ia supernovae
“standard candles”
Permutter
Riess
The expansion of the
Universe is accelerating.

10. Mysteries in the sky
- There seems to be more matter than what we observe…
- The big-bang is big-banging faster than we thought …

11. Mysteries in the sky
- There seems to be more matter than what we observe…
- The big-bang is big-banging faster than we thought …
“dark matter” (but we do not know what it is)
“dark energy” (but we do not know what it is)

12. Mysteries in the sky
Anomalies and tensions in Λ-CDM (Review in [Peebles 2022])
- Baryonic Tully-Fischer rotation curve
- Acceleration of the expansion
- Anomalous abundance of small haloes
- Formation time of structures
- Anomalous dipole
- Anomalous bulk flow
- …

13. Mysteries in the sky
Anomalies and tensions in Λ-CDM (Review in [Peebles 2022])
- Baryonic Tully-Fischer rotation curve
- Acceleration of the expansion
- Anomalous abundance of small haloes
- Formation time of structures
- Anomalous dipole
- Anomalous bulk flow
- …
We need new ideas,
new models,
new equations here !

14. pc/h : parsec (= 3.2 light year)
The millenium simulation project,
Max Planck Institute fur Astrophysik
Simulations

15. Observations

16. 1. Newton 2. Brenier-Monge-Ampère 3. Optimal Transport
4. Discrete Optimal Transp.
5. Large Deviations Pple.
6. The Path Bundle Method

17. 1. Newton-Poisson
M
m
r

18. M
m
r
1. Newton-Poisson

19. M
m
r
1. Newton-Poisson

20. 1. Newton-Poisson

21. 1. Newton-Poisson

22. (F = ma)
Gravity for a set of particles
(N-body)
Lagrangian coordinates
1. Newton-Poisson

23. Gravity for a density field ?
Eulerian coordinates
1. Newton-Poisson

24. Gravity for a density field ?
Eulerian coordinates
(F=ma)
1. Newton-Poisson

25. Gravity for a density field ?
Eulerian coordinates
(F=ma)
1. Newton-Poisson

26. Gravity for a density field ?
Eulerian coordinates
(F=ma)
Green function
1. Newton-Poisson

27. Gravity for a density field ?
Eulerian coordinates
(F=ma)
Green function
1. Newton-Poisson

28. Gravity for a density field ?
Eulerian coordinates
(F=ma)
Velocity field Correction term
(convective derivative)
1. Newton-Poisson

29. Gravity for a density field ?
Eulerian coordinates
(F=ma)
1. Newton-Poisson

30. Gravity for a density field ?
Eulerian coordinates
(F=ma)
(Mass conservation continuity eqn)
1. Newton-Poisson

31. Gravity for a density field ?
Eulerian coordinates
(F=ma)
(Mass conservation continuity eqn)
1. Newton-Poisson

32. 1. Newton 2. Brenier-Monge-Ampère 3. Optimal Transport
4. Discrete Optimal Transp.
5. Large Deviations Pple.
6. The Path Bundle Method

33. 2. Brenier-Monge-Ampère
Taylor expansion of the determinant of a matrix around the identity:

34. 2. Brenier-Monge-Ampère
Taylor expansion of the determinant of a matrix around the identity:

35. 2. Brenier-Monge-Ampère
Taylor expansion of the determinant of a matrix around the identity:

36. 2. Brenier-Monge-Ampère
Taylor expansion of the determinant of a matrix around the identity:

37. 2. Brenier-Monge-Ampère
Taylor expansion of the determinant of a matrix around the identity:

38. 2. Brenier-Monge-Ampère
Taylor expansion of the determinant of a matrix around the identity:

39. Newton-Poisson Brenier-Monge-Ampère

40. 1. Newton 2. Brenier-Monge-Ampère 3. Optimal Transport
4. Discrete Optimal Transp.
5. Large Deviations Pple.
6. The Path Bundle Method

41. 3. Optimal Transport and Monge-Ampère

42. 3. Optimal Transport and Monge-Ampère

43. 3. Optimal Transport and Monge-Ampère

44. 3. Optimal Transport and Monge-Ampère
T(r)

45. 3. Optimal Transport and Monge-Ampère
B

46. 3. Optimal Transport and Monge-Ampère
T-1(B)
B

47. 3. Optimal Transport and Monge-Ampère
T-1(B)
B

48. 3. Optimal Transport and Monge-Ampère
T-1(B)
B

49. 3. Optimal Transport and Monge-Ampère
T-1(B)
B

50. 3. Optimal Transport and Monge-Ampère
T-1(B)
B
Lagrange multiplier associated with
the constraint

51. 3. Optimal Transport and Monge-Ampère

52. 3. Optimal Transport and Monge-Ampère
Optimality conditions

53. 3. Optimal Transport and Monge-Ampère
Optimality conditions

54. 3. Optimal Transport and Monge-Ampère
Optimality conditions

55. 3. Optimal Transport and Monge-Ampère
Optimality conditions

56. 3. Optimal Transport and Monge-Ampère
Optimality conditions
Legendre-Fenchel dual

57. 3. Optimal Transport and Monge-Ampère
Optimality conditions
Legendre-Fenchel dual
Insert into constraint:

58. 3. Optimal Transport and Monge-Ampère
Optimality conditions
Legendre-Fenchel dual
Insert into constraint:
Pointwise:

59. 3. Optimal Transport and Monge-Ampère
Optimality conditions
Legendre-Fenchel dual
Insert into constraint:
Pointwise:
Monge-Ampère equation:

60. 1. Newton 2. Brenier-Monge-Ampère 3. Optimal Transport
4. Discrete Optimal Transp.
5. Large Deviations Pple.
6. The Path Bundle Method

61. 4. Discrete Optimal Transport

62. 4. Discrete Optimal Transport
T(r)

63. 4. Discrete Optimal Transport
N points ri N points qi

64. 4. Discrete Optimal Transport
T(ri) = qσ(i)

65. 4. Discrete Optimal Transport
T(ri) = qσ(i)
σ :The permutation that minimizes

66. 4. Discrete Optimal Transport
T(ri) = qσ(i)
σ :The permutation that minimizes

67. 4. Discrete Optimal Transport
T(ri) = qσ(i)
σ :The permutation that minimizes

68. 1. Newton 2. Brenier-Monge-Ampère 3. Optimal Transport
4. Discrete Optimal Transp.
5. Large Deviations Pple.
6. The Path Bundle Method

69. 5. Large Deviation Principle
σ :The permutation that minimizes

70. 5. Large Deviation Principle
σ :The permutation that minimizes
Why ?
Can we deduce this formula from something else ?

71. 5. Large Deviation Principle
Idea has similarities
with least action

72. 5. Large Deviation Principle
Idea has similarities
with least action
Extremize action between
fixed initial and final conditions.

73. 5. Large Deviation Principle
Idea has similarities
with least action
Extremize action between
fixed initial and final conditions.
Deduce law of motion
(differential relation)

74. 5. Large Deviation Principle
Idea has similarities
with least action
Extremize action between
fixed initial and final conditions.
Deduce law of motion
(differential relation)
Extrapolate it

75. 5. Large Deviation Principle
M indistinguishable particles
Independent Brownian motion
No interaction

76. 5. Large Deviation Principle
We suppose that we observe
them here after T seconds

77. 5. Large Deviation Principle
We suppose that we observe
them here after T seconds
What is the “most probable” motion
that accounts for the observation ?

78. 5. Large Deviation Principle
Probability of observing the
particles here after T seconds:

79. 5. Large Deviation Principle
Probability of observing the
particles here after T seconds:
It’s a soft inf !

80. 5. Large Deviation Principle
Probability of observing the
particles here after T seconds:
Make “temperature” ϵ tend to 0:

81. 5. Large Deviation Principle
Trajectories become geodesics

82. 5. Large Deviation Principle
Along these geodesics:

83. 5. Large Deviation Principle
Along these geodesics:

84. 1. Newton 2. Brenier-Monge-Ampère 3. Optimal Transport
4. Discrete Optimal Transp.
5. Large Deviations Pple.
6. The Path Bundle Method

85. 6. The Path Bundle Method
Initial condition (homogeneous)

86. 6. The Path Bundle Method
Initial condition (homogeneous)
Structures formation

87. 6. The Path Bundle Method
Initial condition (homogeneous)
Structures formation
Observation

88. 6. The Path Bundle Method

89. 6. The Path Bundle Method

90. 6. The Path Bundle Method

91. 6. The Path Bundle Method
: barycenter of

92. 6. The Path Bundle Method

93. 6. The Path Bundle Method

94. Results – Cosmological simulation
• 300 million particles
• 200 Mpc/h
• Λ-CDM initial conditions [Planck]
• Newton-Poisson and BMAG

95. Results – Simulation with 300 M cells

96. Results – Simulation with 300 M cells

97. Results – Simulation with 300 M cells

98. Results – Simulation with 300 M cells

99. Halo masses
Halo shapes
Angular momentum
Rotation curves

100. Results – Conclusions
BMAG is a small non-linear modification of Newtonian dynamics
Differences:
• Larger number of filaments
• Smaller number of small haloes
• Haloes spin faster. Origin of angular momentum of disk galaxies ?
• Centrail density profile of haloes is flatter
• More power on large scales and less power on small scales

101. Results – Conclusions
BMAG is a small non-linear modification of Newtonian dynamics
Differences:
• Larger number of filaments
• Smaller number of small haloes
• Haloes spin faster. Origin of angular momentum of disk galaxies ?
• Centrail density profile of haloes is flatter
• More power on large scales and less power on small scales
Can be falsified with future observational surveys

102. Results – Conclusions
BMAG is a small non-linear modification of Newtonian dynamics
Differences:
• Larger number of filaments
• Smaller number of small haloes
• Haloes spin faster. Origin of angular momentum of disk galaxies ?
• Centrail density profile of haloes is flatter
• More power on large scales and less power on small scales
Can be falsified with future observational surveys
Questions:
•BMAG as the weak field limit of another strong-field theory ?
•BMAG emerging from GR (or other modified theories of gravity) ?
•Entropic gravity ?

103. Large Scale
Structure
3D, Euclidean
L = 1 GPc/h N = 109
A
Future works:
Exploring the shape of
the Universe

104. Large Scale
Structure
3D, Euclidean
Galactic dynamics
6D phase space
General Relativity
4D, Riemannian
Calabi-Yau Manifolds
10D, Complex
Geometric
complexity
Scale
L = Planck
L = 1 Pc/h N = 1 … 10
L = 1 kPc/h N = 106
L = 1 GPc/h N = 109
A
B
C
D
Future works:
Exploring the shape of
the Universe

105. References on Cosmology and OT
Nature 2002, Frisch, Matarrese, Mohayaee, Sobolevski
Geom. & Func. Ana., 2004, Brenier
Confluentes Math, 2011, Brenier
Analysis & PDE, 2023, Ambrosio, Baradat and Brenier
MNRAS 2021, L, Mohayaee, von Hausegger
Physical Review Letters 2021, von Hausegger, L, Mohayaee
Journal of Computational Physics 2022, L
Physical Review Letters 2022, Nikhaktar, Sheth, L, Mohahayee
Physical Review D, 2023, Nikhaktar, Padmanabhan, L, Sheth, Mohayaee
Submitted - Brenier, L, Boldrini, Mohayaee